The Hénon map is a two-dimensional dynamical system introduced by French mathematician Michel Hénon in 1976. It is a simple, non-linear system used to study chaos theory and is often used as a model of chaotic behaviour in physical systems.
Two equations define the Hénon map:
xn+1 = 1 - ax2 + yn
yn+1 = bxn
Where a and b are constants.
The Hénon map is an example of a discrete-time dynamical system, which means that the values of x and y are calculated at regular intervals. The map is iterated, meaning that the output of each iteration is used as the input for the next iteration.
The Hénon map can be visualized as a graph of x and y values. When the constants a and b are set to specific values, the graph will often show a pattern of chaotic behaviour. This behaviour is characterized by a sensitive dependence on initial conditions, meaning that small changes in the starting values of x and y can lead to large changes in the output of the map.
The Hénon map is often used to study chaos theory, which is the study of complex systems that show unpredictable behaviour. It has been used to model systems such as the weather, population growth, and stock markets.
The Hénon map is also used in fields such as cryptography and robotics, where it can be used to generate random numbers or to control robotic systems.
Overall, the Hénon map is an important tool in studying chaos theory and has applications in many different fields.
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